Incorporating Directionality in Transversal-Resonator-Based Bandpass Filters With Tunable Transfer Function Characteristics

This paper introduces a detailed methodology for the realization of multi-functional transversal resonator-based non-reciprocal bandpass filters (NR-BPFs) that combine the functionality of a single-/multi-band BPF and an RF isolator. Directionality is achieved through spatiotemporal modulation (STM) whereas transfer function reconfigurability is obtained by only tuning the resonant frequency of its constituent resonators. A detailed design methodology to synthesize the response of transversal resonator-based STM arrays is introduced in this work for the first time, facilitating the synthesis of advanced RF filtering transfer functions based of transversal resonator arrays with incorporated directionality. The operating principles and scalability of the design method are demonstrated through the analysis of four distinct transversal resonator-based STM filtering topologies that facilitate the realization of high-order and highly-modular single-band and multi-band transfer functions with multiple levels of RF tuning including frequency tuning, bandwidth tuning, band controllability, and intrinsic switch-off capabilities. The concept has been validated at UHF band through the manufacturing and testing of four lumped-element NR-BPFs.

Abstract-This paper introduces a detailed methodology for the realization of multi-functional transversal resonator-based non-reciprocal bandpass filters (NR-BPFs) that combine the functionality of a single-/multi-band BPF and an RF isolator.Directionality is achieved through spatiotemporal modulation (STM) whereas transfer function reconfigurability is obtained by only tuning the resonant frequency of its constituent resonators.A detailed design methodology to synthesize the response of transversal resonator-based STM arrays is introduced in this work for the first time, facilitating the synthesis of advanced RF filtering transfer functions based of transversal resonator arrays with incorporated directionality.The operating principles and scalability of the design method are demonstrated through the analysis of four distinct transversal resonator-based STM filtering topologies that facilitate the realization of high-order and highly-modular single-band and multi-band transfer functions with multiple levels of RF tuning including frequency tuning, bandwidth tuning, band controllability, and intrinsic switch-off capabilities.The concept has been validated at UHF band through the manufacturing and testing of four lumpedelement NR-BPFs.

I. INTRODUCTION
E MERGING wireless communication systems are increas- ingly in need of highly-reconfigurable multi-functional RF components to enhance the link performance and enable challenging applications such as full-duplex spectrum usage or joined sensing and communications (JCAS).In this regard, non-reciprocal devices such as RF isolators and circulators are essential for isolating different parts of their RF front-ends (e.g., separating transmit from receive paths) and cancelling inter-stage reflections, with the aim of improving the signalto-noise (SNR) ratio [1], [2].Furthermore, reconfigurability alongside RF filtering is also needed in the RF front-ends to ensure adaptability to dynamically changing spectral conditions and to provide robust interference suppression.
This paper presents a comprehensive design methodology and a practical validation concept for transversal-resonatorbased NR-BPF with widely-tunable transfer function characteristics, high levels of isolation between the forward and the backward direction and multiple levels of single and multi-band transfer function tuning.Specifically, the manuscript demonstrates for the first time how to model transversal STM resonator-based arrays that in turn facilitate the synthesis of advanced RF filtering transfer functions with incorporated directionality.
The operating principles of the transversal-resonator-based NR-BPF concept and its scalability to highly-selective transfer functions are discussed through its application to four distinct high order filtering topologies.They include a single-band BPF based on two cascaded transversal cells (Topology 1), a single-band fourth-order box-like NR-BPF (Topology 2), a single-band sixth-order box-like NR-BPF (Topology 3), and a six-resonator-based NR-BPF using three series transversal cells (Topology 4), which exhibits multiple levels of transfer function tunability alongside a dual-band response.Multilevels of RF tuning can be obtained for the four topologies using a simple RF tuning mechanism where only the resonant frequencies of the resonators are altered.These include center frequency tuning, bandwidth (BW) tuning and intrinsic switching off for all the four topologies, as well as flexible band controllability for Topology 4. For all of the aforementioned tuning states, high levels of IS (> 20 dB) between the forward and the reverse direction can be obtained.When compared to a conventional solution where an amplifier is cascaded in series with a tunable BPF to incorporate non-reciprocity alongside amplification, the STM-based NR-BPFs of this work have significantly smaller size.They don't consume any power, nor generate any heating nor require inter-stage matching networks or harmonic terminations.Furthermore, they allow for dynamic reconfiguration of the direction of propagation.
The manuscript is organized as follows.Section II establishes the theoretical foundations for the transversal resonator-based concept and demonstrates how STM doublets can be modelled using a simple synthesis method.Whereas spectral network parameter matrices of a time-varying resonator have been developed in [30], and applied in in-line BPF configurations [13], this method has not yet been applied to STM transversal cells, which serve as the foundational counterpart for a wide variety of filters.In Section III, a comprehensive design procedure is discussed for selecting the STM parameters for these filters with the purpose of achieving high directivity.The experimental validation of the concept for all four filtering topologies is discussed in Section IV.Finally, Section V highlights the main contributions of this work.

II. ANALYSIS OF SPATIOTEMPORALLY-MODULATED TRANSVERSAL RESONATOR-BASED BPFS
The details of the STM two-resonator transversal cell or doublet are illustrated in Fig. 1 in terms of its coupling routing diagram (CRD) [Fig.1(a)] and its circuit equivalent [Fig.1(b)].The doublet, denoted by its ABCD matrix N t , can be considered as the parallel connection of two networks marked in blue and red, with their respective ABCD matrices being N 1 and N 11 .The red (blue) network comprises a cascaded connection of an input inverter M 1,2 (M 1,3 ), an STM resonator R 1 (R 2 ) and an output inverter M 2,4 (M 3,4 ).The circuit equivalent of the STM doublet is provided in Fig. 1 (b), where the STM resonators are implemented as parallel LC tanks whose capacitors are time-variant with capacitance C 0x (t) as given in (1) and their inductors L x (x is the index number of the resonator R x ) are time invariant.C 0x is the nominal (dc) capacitance, ξ x is the modulation depth, f m is the modulation frequency, and φ x is the phase of the modulating waveform of the resonator R x .
The spectral admittance matrices Y Cx and Y Lx for the shunt time-varying capacitor and the time-invariant inductor are provided in (2), as shown at the bottom of the next page.These matrices incorporate the energy conversion between different intermodulation frequencies f ±n f m (n = 0, 1, 2) [13], [30].As such, they should have theoretically infinite dimensions (i.e., n is infinite), but only a limited number of intermodulation frequencies need to be considered in practice.Using the expressions in (1) and (2), the ABCD matrix of a single time-varying resonator N Rx and a time-invariant inverter N Ma,b can be calculated using (3) and (4), where a and b are the indices of the inverters, U is the unity matrix and 0 is the zero matrix with the same dimension as the spectral admittance matrices Y Cx and Y Lx .Since the blue and the red parts are the cascade connection of three two-port networks, N 1 and N 11 can be easily found by multiplying the ABCD matrices of the individual two-port subnetworks as shown in ( 5) and (6), respectively [31].N t can be derived from N 1 and N 11 by considering the signal relations in ( 7)- (10) and its matrix elements are provided in ( 11)- (14).They have been obtained by substituting (7) and (8) in the relationships outlined in ( 9) and ( 10) that lead to V t1 , V t2 , I t1 and I t2 , which can be used to calculate the ABCD elements of N t .
The S-parameters of the transversal cell S t can be calculated from N t using (15), as shown at the bottom of the page, where S 11 , S 21 , S 12 and S 22 represent the elements in S t and A, B, C and D are the elements of N t as provided in [13].
Once the ABCD and scattering matrices of a single STM two-resonator transversal cell are obtained, the scattering matrices of filtering topologies based on STM transversal cells becomes a straightforward process, as they can be regarded as the cascading connection between single STM resonators, time-invariant inverters and STM transversal cells.
To evaluate the feasibility of the aforementioned design method for the synthesis of highly-selective filtering transfer functions, four fully-directional filtering topologies (Topology 1-4) are considered.Specifically, Topologies 1 and 2 demonstrate two different ways to realize a 4 th order transfer functions by arranging 4 STM resonators in either two cascaded doublets or in a box-like configuration.Furthermore, by adding STM resonators at the input and output ports of the Topology 2 CRD, a six th order transfer function can be obtained as shown in Topology 3. To facilitate multiple levels of tuning within a single RF filtering configuration, three series-cascaded transversal cells are used as shown in the Topology 4 CRD.

A. Topology 1
The CRD details of the Topology 1 fully-directional STMbased BPF are provided in Fig. 2. In each transversal cell, the two resonators are asynchronously tuned and the inverters Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
M 3,4 and M 5,7 need to have opposite polarity to obtain a four-pole bandpass transfer function.All the STM resonators (denoted by the black circles) consist of a parallel time-invariant inductor with the same inductance L r and timevarying capacitors, whose capacitance are labeled in the boxes next to them.Specifically, all the time-varying capacitors have a nominal capacitance C 0x , and have the same modulation frequency f m and modulation depth ξ .To enable directionality, the phase arrangement for the capacitors should follow the configuration labeled in the figure, i.e., 0 • for R 2 and R 3 and ϕ • for R 6 and R 7 .For analysis purposes, the Topology 1 S-parameters can be calculated from its overall ABCD matrix that is equal to N t1 × N M4,5 × N t2 where N t1 is the ABCD matrix of the first doublet, N M4,5 is the ABCD matrix of the inverter and N t2 is the ABCD matrix of the last doublet.
For validation purposes, the synthesized S-parameters of Topology 1 are compared with their corresponding circuit-simulated ones (obtained in ADS from Keysight, utilizing ideal LC components and non-linear capacitors with the time-varying capacitance) and are provided in Figs.2(b) and 2(c) using the element values in the caption of Fig. 2 and considering an intermodulation index of 4 (intermodulation frequencies f ±nf m (n = 0, 1, 2, 3, 4) is considered).It should be noted that the coupling element values have been obtained by optimization.As shown, the BPF works at 700 MHz with a fractional BW of 10%.Furthermore, as observed in Figs.2(b), when STM is OFF (ξ = 0), the filter response is reciprocal, i.e., |S 21 | = |S 12 |.When STM is ON (ξ ̸ = 0), really high IS can be created between the forward and backward transmission directions, as shown in Fig. 2(c).Moreover, a strong agreement between the synthesized results and the circuit simulated results can be observed, indicating the validity of the proposed synthesis method using STM transversal cells.
The tuning capabilities of Topology 1 are examined in Fig. 3 through various design cases.As shown in Fig. 3(a), the center frequency of the filter can be tuned to lower (higher) frequencies by increasing (decreasing) the capacitance of all the four resonators at the same time whilst preserving high isolation between the forward and the backward direction.The BW tuning capabilities of the NR-BPF are presented in Fig. 3(b).Wider (narrower) BW can be achieved by increasing (decreasing) the value of C 02 and C 06 while decreasing (increasing) the value of C 03 and C 07 .Furthermore, as shown in Case 4 in Fig. 3(b), the BPF can be intrinsically-switched off when C 02 , C 06 , C 03 and C 07 have the same value.For all of these cases the component element values are provided in the caption of Figs. 2, 3.It is noted that the parameters of the time-varying capacitors should be adjusted accordingly in different tuning stages to achieve optimal isolation levels.

B. Topology 2
The details of the Topology 2 CRD are provided in Fig. 4. It demonstrates an alternative way to functionalize a fourth order transfer function by connecting two synchronously tuned resonators R 2 and R 5 , directly to a single transversal cell, forming a box-like topology.A single-band bandpass transfer function can be obtained when resonators R 2 and R 5 are set to resonate at the center frequency of the BPF, while R 3 and R 4 resonate at lower and higher frequencies.Using as a basis the design methodology mentioned above, a fourth order BPF was designed at 700 MHz with a fractional BW of 9% using the parameters specified in the captions of Fig. 4 (coupling element values have been obtained by optimization).Furthermore, R 2 is modulated with an AC signal having a phase of 0 • , R 3 and R 4 with an AC signal having a phase of ϕ • and R 5 with a phase of 2ϕ • .Figs. 4(b) and 4(c) demonstrate a comparison between the synthesized S-parameters and the ones obtained by circuit simulations which appear to be in excellent agreement.The tuning capabilities of Topology 2 are shown in Fig. 5 in terms of center frequency (Fig. 5(a), Case 1-3), BW tuning (Fig. 5(b), Case 1-3)) and intrinsic switching off (Fig. 5(b), Case.4).They were obtained in a similar manner to the ones of Topology 1.

C. Topology 3
The Topology 3 CRD in Fig. 6 demonstrates scalability to higher-order transfer functions by cascading two additional resonators in the box-like Topology 2 configuration, resulting in a sixth-order single-band bandpass transfer function.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.As depicted in Fig. 6(a), to achieve directionality, the modulating AC signals need to be applied with the following phase progression: 0 • for the signal applied to R 2 , ϕ • for R 3 , 2ϕ • for R 4 and R 5 , 3ϕ • for R 6 , and 4ϕ • for R 7 .The comparison between the synthesized S-parameters and the circuit-simulated ones is presented in Figs.6(b) and 6(c).They have been obtained with the same method as above and appear to be in an excellent agreement, successfully validating the proposed synthesis method.The tuning capabilities of Topology 3 are similar to the ones obtained for Topology 2, thus omitted here for brevity.

D. Topology 4
Topology 4 facilitates the realization of multi-functional/ configurable transfer functions by only tuning the resonant frequencies of its constituent resonators.Specifically, it enables various levels of tunability, such as center frequency, bandwidth, intrinsic switching off, reconfigurability of the number of passbands through band merging and cancellation and tuning of the out-of-band rejection by transmission zero (TZ) reallocation.To achieve these functionalities, all inverters share the same polarity, and the phases of the two resonators within the same STM transversal cell are identical.However, a successive phase difference is required for adjacent transversal cells, as illustrated in Fig. 7.
A summary of the tuning capabilities of Topology 4 is provided in Fig. 8, using the component element values in the caption and in Table I.The coupling coefficients for all 6 tuning states are the same as listed in Fig. 7.As shown in Fig. 8(a), a third order dual-band transfer function can be generated when resonators R 2 , R 6 and R 10 are set to the same frequency that coincides with that of the higher passband while resonators R 3 , R 7 and R 11 resonate at the lower passband.In another configuration [see Case 2 in Fig. 8(b)],

TABLE I PARAMETERS FOR TIME-VARYING CAPACITANCE FOR
THE DIFFERENT TUNING CASES OF FIG. 8 an equi-ripple response can be generated by setting all the resonators to resonate at the same frequency.Furthermore, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.show the transfer functions that can be obtained when placing resonators R 3 , R 6 , R 7 and R 11 inside the passband, facilitating quasi-elliptic transfer functions with two symmetrically allocated TZs around the passband or with just one TZ.The filter can also be intrinsically switched-off [see Case 6 in Fig. 8(f)] when setting resonators R 6 and R 7 to resonate at the same frequency while offsetting the other resonators from the center frequency of the filter.Although not shown here, the center frequency in all the aforementioned cases can be freely tuned through altering all the resonators' resonating frequency at the same time as it will be shown in the experimental validation of this concept.Furthermore, the zeros in Cases 3 to 5 can be placed at either side of the passband.Their location can be altered by tuning the corresponding resonators.As can be observed from Fig. 8, large isolation levels can be achieved in all the tuning states, demonstrating the effectiveness and versatility of the proposed topologies.

A. STM Parameters Selection
Directionality in the transversal-based BPFs can be achieved when the capacitance of the constituent resonators follows the sinusoidal variation described in (1).Depending on the selected modulation parameters ξ , f m and ϕ, alternative S-parameter profiles can be obtained.Thus, parametric studies need to be performed to determine the desired transfer function characteristics as shown for the example cases of Topology 1 and Topology 4 in Figs. 9 and 10.Similar studies can be performed for the rest of the filtering topologies, but they have been omitted for brevity.In order to effectively present the parametric sweeping results, two data visualization methods have been adopted, namely line plots and heatmaps.These two complementary data visualization techniques provide a comprehensive view of the parametric sweeping results, and their characteristics are summarized as follows: 1. Line plots [Figs.Using as a basis these performance visualization methods, the Topology 1 performance trade-off metrics are provided in Fig. 9. Specifically, a f m = 30 MHz results in the highest D and an acceptable insertion loss (IL).Larger f m values lead to lower IL but also lower D, while too small f m values lead to high IL and low D as well.Regarding the modulation index ξ , the optimal value is chosen when ξ = 0.07.Although lower ξ values yield lower IL, the D is also lower.Too high ξ values cause both lower D and higher IL.Finally, an optimal value of 60 • can be observed for ϕ, which achieves high directivity with low IL while maintain good transfer function shapes.Fig. 10 illustrates the trade-offs analysis study for Topology 4 when operating in a dual-band mode of operation.The optimal modulation parameters of the dual-band BPF are chosen as f m = 20 MHz, ξ = 0.05 and ϕ = 65 • for high D for both bands while maintain an acceptable IL and good transfer function shapes.Using as a reference these plots, optimal STM parameters can be selected for the desired S-parameters for Topology 1 and 4 and are demonstrated in Figs.3(b) and 3(h), respectively.

B. Practical Realization Aspects
1) Lumped-Element-Based Implementation Method: Based on the CRDs and the coupling coefficients provided in the previous sections, the four multi-functional filtering topologies can be implemented using LEs and conventional filter design techniques to specify the LE element values as for example the ones described in [32].Next, the DC biasing on the varactor is selected so that it gives the desired resonator capacitance.Subsequently, the AC modulation parameters f m , V M , and φ are obtained through parametric simulations according to the desired IL in the forward direction and the desired IS in the reverse direction.Lastly, the tuning capabilities are obtained by altering the DC and AC biasing.
Fig. 11 shows a design example when implementing the CRD of Topology 2 in Fig. 4(a).Firstly, a reciprocal filter is designed based on the CRD and the coupling coefficients and the equivalent circuit is given in Fig. 11(a), where C xr (x = 2,3,4,5) is the capacitance L r is the inductance of the resonators, and J a,b is the inverter value after impedance scaling and bandpass transformation to a center frequency of ω 0 and a fractional bandwidth of FBW.The values can be calculated using (16).The inductance of the resonators are set to the same value of L r in our designing cases.After acquiring all the parameters of the equivalent circuit in Fig. 11(a), the STM resonators and the static inverters can be implemented using lumped elements as shown in Figs.11(b) and (c), respectively.Specifically, a parallel inductor L r , a static capacitor C r and a varactor diode D v constitute the inductance and capacitance in each resonator.DC biasing V DC is applied through a 10 M resistor R DC to set the nominal resonant frequency of each resonator.All the resonators are modulated by low frequency AC signals with frequency f m , amplitude V m and phase φ so that the capacitance of the varactor satisfies the sinusoidal variation as the AC signal applied to it.A low-pass filter is added after each AC source to allow for the modulating AC signals to be applied on the resonators while protecting the AC source from the high frequency RF signals.C DC1 and C DC2 are used as DC blocks to prevent the DC signals from entering the AC sources and the RF circuit.Positive coupling coefficients are materialized with first order low-pass pi-networks whereas the negative ones with first order high-pass networks as shown in Fig. 11(c) using (17).To reduce the number of circuit elements required, the parallel capacitors of the inverters are combined with the ones of the adjacent resonators.
2) Sensitivity Analysis: To evaluate the effect of tolerances on the performance of the non-reciprocal RF filtering components, a sensitivity analysis can be conducted for the tolerances which are typically 2%.For the sake of illustration, the circuit architectures of Topology 1 and Topology 2 are analysed through Monte Carlo simulations.The components marked in yellow in Figs.12(a) and 13(a) are meant to be static and as such are chosen to have a 2% tolerance.The simulated results for Topology 1 and Topology 2 are presented in Fig. 12(b) and 13(b), respectively.The solid lines denote the simulation results when all the components have the nominal values while the shaded ones indicate how the S parameters vary with a 2% tolerance for the chosen components in 1600 iterations.It should be noted that for all the 1600 iterations, the same modulation parameters f m , ξ and ϕ are adopted.As shown, the effect of the tolerances is almost unnoticeable with the return loss being always below 10 dB and isolation levels remaining >20 dB when STM is turned on.

IV. EXPERIMENTAL VALIDATION
For proof of concept validation purposes of the transversal resonator-based NR-BPF concept, four filter prototypes (Filters 1-4) implementing the CRD Topologies 1-4 were designed, and manufactured on a RO4003C dielectric substrate with a height of 1.5 mm.The S-parameters for the four filters, including their tuning capabilities, were characterized using a Keysight PNA-X N5244A network analyzer.

A. Filter 1
The photograph of the manufactured prototype and the utilized lumped element components are provided in Fig. 14.A comparison of the RF measured and EM simulated S-parameters for one tuning state is shown in Fig. 15.They appear to be in a good agreement successfully validating the proposed concept.The measured 20 dB IS BW (the frequency range when IS> 20 dB) is 647.5 MHz to 659.5 MHz (12 MHz).The frequency band is marked in light purple.The RF measured tuning capabilities for Filter 1 are presented in Fig. 16.Although a finite number of tuning states is shown, the filter exhibits continuous tuning range.Specifically, its center frequency can be tuned between 571 MHz to 724 MHz (1.27) with an insertion loss (IL) ranging from 4.8 dB to 6.5 dB and isolation (IS) between the forward and the backward direction from 21 dB to 33 dB.Fig. 16(b) demonstrates the BW tuning characteristics of the filter which exhibits a tunable BW between 39.5 MHz to 73.9 MHz (1.87) with IL ranging from 5.9 dB to 6.8 dB and IS between its forward and its backward direction between 25 dB to 35 dB.The intrinsic switch-off capability of Filter 1 is also demonstrated in Fig. 16 (b) and is represented by the green traces in Case 4.

B. Filter 2
Fig. 17 depicts the photograph of the manufactured prototype of Filter 2 and a comparison between the RF-measured and the EM-simulated S-parameters for one tuning state are presented in Fig. 18.For this specific tuning state, the measured 20 dB IS BW is 667 MHz to 705 MHz (38.4 MHz).As can be seen from Fig. 19(a), the center frequency of Filter 2 can be tuned between 630 MHz to 735 MHz (1.17).For these states IL varies between 4.8 dB to 6.7 dB and the IS between the forward and the backward direction between 20 dB and 30 dB.Continuous BW tuning between 24.8 MHz to 58.1 MHz (2.34) is shown in Fig. 19(b).For these states, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

C. Filter 3
The experimental prototype of Filter 3 is provided in Fig. 20.Fig. 21 showcases the good agreement between its EM simulated and RF measured performance for one tuning state further supporting the effectiveness of the transversal resonator-based NR-BPF concept.For this tuning state in Fig. 21, the measured 18 dB IS BW is 670 MHz to 718 MHz (48 MHz).As shown in Fig. 22(a), Filter 3 allows for center     4.5 dB and 5.7 dB and the IS between the forward and the backward direction between 30 dB and 45 dB.As illustrated in Fig. 22(b), the BW tuning range of Filter 3 was measured between 52.9 MHz and 84 MHz (1.57), with IL ranging from 4.9 MHZ to 6.7 dB and IS from 27 dB to 34 dB.Filter 3 can also be intrinsically switched-off as demonstrated in Case 4 in Fig. 22.

D. Filter 4
Fig. 23 shows the manufactured prototype of the multi-configurable Filter 4. Figs.24(a) and 24(b) provide the comparison between the RF-measured and the EMsimulated S-parameters when operating at its dual-band and its    sign of the phase progression of the resonators in each band, the direction of propagation can be tuned as shown in Fig. 27   provided in Table II.As shown, Filters 1, 2, and 3 showcase competitive performance in terms of tuning range, isolation, and return loss when compared to the state-of-the-art.Filters 1 to 3 also exhibit higher order when compared to many existing STM-based non-reciprocal BPFs [13], [16], [17], [18], [20], [21], [22], [23], [24], having higher selectivity and out-ofband rejection.Moreover, Filter 4 shows the highest levels of reconfigurability among all the existing NR-BPFs.It can support both single-and dual-band transfer functions and offers independent tuning of two passbands.Moreover, it can reconfigure its out-of-band response with various TZ configurations, providing improved flexibility in adapting to different system requirements and operating conditions.Furthermore, the proposed filters can be intrinsically switched-off, and can be exploited to reduce the number of RF switches in the RF front-end.

V. CONCLUSION
This paper reported on a comprehensive design methodology for reconfigurable NR-BPFs using STM transversal cells.The proposed method was applied to the design of four filter topologies and demonstrated the potential to achieve single-and multi-band non-reciprocal transfer functions with enhanced reconfigurability.Four proof-of-concept prototypes were manufactured and measured and validated the effectiveness of the transversal cell-based NR-BPF concept in achieving wider tuning ranges, high isolation levels, and superior reconfigurability compared to state-of-the-art STM-based non-reciprocal BPFs.These results underline the potential of using STM transversal cells in designing multi-functional RF components for emerging RF communication systems.

Fig. 1 .
Fig. 1.STM doublet concept.(a) Coupling routing diagram.Colored circles with arrows: STM resonators, white circles: non-resonating nodes.(b) Equivalent circuit where the STM resonators are representd by time-varying capacitors and time-invariant inverters and inductors.N t , N 1 and N 11 are the ABCD matrices of the doublet, the upper transversal path and the bottom transversal path.
9(a) and 10(a)]: In these figures, the amplitudes of the S-parameters (|S 11 |, |S 21 |, and |S 12 |) along with the directivity (D = |S 21 |-|S 21 |) are plotted on the y-axis against frequency on the x-axis.Curves representing different modulation parameters are plotted in the same coordinate

Fig. 10 .
Fig. 10.S-parameters of Topology 4 operating in the dual band mode when varying f m , ξ and ϕ.(a) Line plots.(b) Heatmap plots.system using varying colors, indicated by the color bar on the right side of each row.This visualization method allows for convenient observation of how the S-parameter shapes change with the modulation parameters.2. Heatmaps [Figs.9(b) and 10(b)]: These figures employ a heatmap approach to display the S-parameters as colors in a two-dimensional matrix.One axis represents the frequency range, while the other axis represents the modulation parameters.The color intensity in each cell indicates the value of the corresponding S-parameter.Using as a basis these performance visualization methods, the Topology 1 performance trade-off metrics are provided in Fig.9.Specifically, a f m = 30 MHz results in the highest D and an acceptable insertion loss (IL).Larger f m values lead to lower IL but also lower D, while too small f m values lead to high IL and low D as well.Regarding the modulation index ξ , the optimal value is chosen when ξ = 0.07.Although lower ξ values yield lower IL, the D is also lower.Too high ξ values cause both lower D and higher IL.Finally, an optimal

Fig. 12 .
Fig. 12. Sensitivity analysis of Topology 1.(a) Detailed circuit schematic.(b) Monte Carlo simulation results with 1600 iterations and 2% tolerance for the static components marked in yellow when STM is ON.

Fig. 13 .
Fig. 13.Sensitivity analysis of Topology 2. (a) Detailed circuit schematic.(b) Monte Carlo simulation results with 1600 iterations and 2% tolerance for the static components marked when STM is ON.

Fig.
Fig. Manufactured prototype of Filter 2 based on the Topology 2 CRD.

Fig. 18 .
Fig. 18.Comparison of RF measured and EM-simulated S-parameters for one tuning state of Filter 2. The purple band marks the frequency range when |S 12 | <−20 dB.

Fig. 20 .
Fig. 20.Manufactured prototype of Filter 3 based on the Topology 3 CRD.frequency tuning between 655 MHz and 760 MHz (1.16).For these states, the in-band passband IL varies between

Fig. 21 .
Fig. 21.Comparison of RF measured and EM-simulated S-parameters for Filter 3. The purple band marks the frequency range when |S 12 | <−18 dB.

Fig. 26 .
Fig. 26.RF measured f cen tuning capabilities of Filter 4 when operating at its single-band mode.

Fig. 27 .
Fig. 27.RF measured reconfigurable capabilities of Filter 4 when operating at its dual-band mode.(a) Both bands transmit in the same direction.(b) Each band is transmitting in a different direction.
(a)  and (b) by altering θ from 65 • to −65 • .Furthermore, Fig.28(a) and (b) show how the two passbands can be tuned independently.Specifically, the lower passband can be tuned between 600-680 MHz with IL between 5.6-6 dB and IS between 23-26 dB.The upper band can be tuned between 716-831 MHz with IL between 4.8-6.2dB and IS between 25-31 dB.Furthermore, Filter 4 can be intrinsically switched-off as demonstrated in Fig.29.Return loss for all prototypes is better than 10 dB throughout the entire tuning range.A comparison of the transversal-cell-based NR-BPF concept with state-of-the-art STM-based non-reciprocal BPFs is

Fig. 28 .
Fig. 28.RF measured f cen tuning capabilities of Filter 4 when operating at its dual-band mode of operation.(a) Upper band f cen tuning.(b) Lower band f cen tuning.
Graduate Student Member, IEEE, and Dimitra Psychogiou , Senior Member, IEEE

TABLE II COMPARISON
WITH THE STATE-OF-ART NON-RECIPROCAL FILTERS